root ofx4-x=10 in rapson matheod

Question

Positive root

Answer 1

x⁴-x=10.

Let f(x)=x⁴-x-10, f'(x)=4x³-1.

Use iteration formula: x_n+1=x_n-f(x_n)/f'(x_n).

We need to find a suitable value for x₀.

f(1)=-10 and f(2)=2 so there's a positive root between x=1 and x=2, therefore let x₀=1.

x₁=1-(-10)/3=13/3, x₂=3.29..., x₃=2.098..., x₄=1.8956..., x₅=1.895688..., x₆=1.85558..., x₇=1.855584529, x₈=x₇ so x=1.855584529 is an approximate positive zero. A more accurate value is 1.85558452864.

We could have used x₀=2 and reached the same accuracy at x₄.